Question

If $\hat{i},\hat{j}$ and $\hat{k}$ represents unit vectors along the $x,y$ and $z-$axes respectively, then find the value of angle $\theta$ between the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}$.

Answer 1

Consider the vector

$\begin{array}{c}\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}\\ \overrightarrow{B}=\hat{i}+\hat{j}\\ \overrightarrow{A}\cdot \overrightarrow{B}=AB\cos \theta \\ \Rightarrow \cos \theta =\frac{\overrightarrow{A}.\overrightarrow{B}}{A}\\ =\frac{\hat{i}\cdot \hat{i}+\hat{j}\cdot \hat{j}}{\sqrt{3}\sqrt{2}}\\ =\frac{1+1}{\sqrt{6}}\\ =\sqrt{\frac{2}{3}}\\ \Rightarrow \cos \theta =\sqrt{\frac{2}{3}}\\ \Rightarrow \theta ={\cos }^{-1}\sqrt{\frac{2}{3}}\end{array}$